Difference between revisions of "Functional derivative"
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that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204016.png" />. | that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042040/f04204016.png" />. | ||
− | In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the [[Gâteaux derivative|Gâteaux derivative]] and the [[Fréchet derivative|Fréchet derivative]]. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see [[Variational calculus, numerical methods of|Variational calculus, numerical methods of]]). | + | In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the [[Gâteaux derivative|Gâteaux derivative]] and the [[Fréchet derivative|Fréchet derivative]]. But the concept of a functional derivative has been applied with success in numerical methods of the classical [[calculus of variations]] (see [[Variational calculus, numerical methods of|Variational calculus, numerical methods of]]). |
Revision as of 22:37, 22 November 2014
Volterra derivative
One of the first concepts of a derivative in an infinite-dimensional space. Let be some functional of a continuous function of one variable
; let
be some interior point of the segment
; let
, where the variation
is different from zero in a small neighbourhood
of
; and let
. The limit
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assuming that it exists, is called the functional derivative of and is denoted by
. For example, for the simplest functional of the classical calculus of variations,
![]() |
the functional derivative has the form
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that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of .
In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the Gâteaux derivative and the Fréchet derivative. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see Variational calculus, numerical methods of).
Comments
The existence of the functional derivative of at
and
apparently means that the Fréchet derivative
of
at
, which is a continuous linear form on the space of admissible infinitesimal variations
, is of the form
for some continuous function
, so that it can be continuously extended to
the
-function at
. In the example this happens only if
is twice continuously differentiable.
Functional derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_derivative&oldid=34863